Question 350476
Please help me solve this equation: Consider the function {{{ f(x)= root(3,x) }}}, {{{ 1<=x<=8 }}}. 

a) Find the average value of f on the interval [1, 8].
For the average value on this interval you will use:
{{{(int(root(3, x), dx, 1, 8))/(8-1)}}}
Writing the radcial in fractional exponent form will probably help with finding the integral:
{{{(int(x^(1/3), dx, 1, 8))/(8-1)}}}
The indefinite integral of {{{x^(1/3)dx}}} is {{{x^(4/3)/(4/3) + C = (3x^(4/3))/4 + C}}}. The definite integral, from 1 to 8, will be:
{{{(3(8)^(4/3))/4 - (3(1)^(4/3))/4}}}
{{{8^(4/3) = 8^((1/3)*4) = (8^(1/3))^4 = 2^4 = 16}}}. So now we have
{{{(3(16))/4 - 3/4}}}
{{{48/4 - 3/4}}}
{{{45/4}}}
Now the average value is:
{{{(45/4)/7 = 45/28}}}<br>
b) Find c such that fave. = f (c).
So we solve:
{{{f(c) = 45/28}}}
{{{root(3, c) = 45/28}}}
Cube both sides:
{{{c = 91125/21952}}}<br>
c) Sketch the graph of f and a rectangle whose area is the same as the area under the graph of f.
The graph of f (without the limited domain) looks like:
{{{graph(400, 400, -1, 11, -1, 5, x^(1/3))}}}
Note:<ul><li>Truncate this graph so that only the part between x = 1 and x = 8 shows.</li><li>I think the intention is for the graph of the rectangle to be on the same graph. Specifically, the bottom of the rectangle should be the on the x axis between 1 and 8 and the sides will be 45/28 in height extending up from the x axis (at x = 1 and x = 8). If done well, the top of the rectangle should intersect the graph of f at x = c =  91125/21952 (which is approximately 4.15)</li></ul>